1) Dollar bills. In a bag are 26 bills. If you take out 20 bills from the bag at random, you have at least one 1-dollar bill, two 2-dollar bills, and five 5-dollar bills. How much money was in the bag?

2) Yin and Yang. The Yin-Yang symbol is based on the figure below, bordered by three semi-circles. How can you divide this shape into two identical shapes?

3) Huge pie. A huge pie is divided among 100 guests. The first guest gets 1% of the pie. The second guest gets 2% of the remaining part. The third guest gets 3% of the rest, etc. The last guest gets 100% of the last part. Who gets the biggest piece?

Solutions:

1) The answer is 78 dollars. Let the number of 1-dollar bills in the bag be a, of 2-dollar bills be b,and of 3-dollar bills be c. We are told that if you take out 20 bills there are 6 left. So we know that

a ≥ 6 + 1 = 7

b ≥ 6 + 2 = 8

c ≥ 6 + 5 = 11

so a + b + c ≥ 26. Since there are only 26 bills in the bag, a + b + c must be equal 26, and so a = 7, b = 8 and c = 11. The total value is (7 x 1) + (8 x 2) + (11 x 5) = 78.

2) I’ve added the grid lines to show exactly where the semicircular cut must go.

When it is the turn of guest number k, a certain part, x, of the pie is left. Guest number k gets x multiplied by (k/100), which is the formula for k per cent of x. And the amount of pie left is is x – x(k/100) = x (100–k)/100.

When it comes to the turn of the next guest, that’s guest k + 1, they will get x multiplied by ((100–k)/100)((k + 1)/100). Therefore guest k + 1 gets more than guest k if (100–k)/100)((k + 1)/100) > k/100. This cancels out and rearranges to the inequality – k2 – k + 100 > 0 , which is true when k = 1, 2, 3, 4, 5, 6, 7, 8 or 9. In other words, the amount of pie taken by each guest keeps on growing until the 9th guest has taken their amount of pie, but that from the 10th guest the subsequent piece is smaller. We can conclude therefore that guest number 10 gets the largest part.

I hope you enjoyed today’s puzzles and I’ll be back in two weeks.

I set a puzzle here every two weeks on a Monday. Send me your email if you want me to alert you each time I post a new one. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

Football School, which I which I co-wrote with Ben Lyttleton, is a book for 7 to 13-year olds children that uses football to explain subjects like English, maths, physics, geography, philosophy and zoology. You (by which I mean any 7-13-year-olds you may know) can check out the Football School YouTube channel, in which Ben and I answer all questions about football and life. Submit your questions and subscribe!